2023/12/12 by Filippo Boni, Boni, Filippo, Simone Dovetta +3
Physics and Astronomy · Engineering · #Quantum chaos and dynamical systems #Lightning and Electromagnetic Phenomena #Electromagnetic Simulation and Numerical Methods
paper · pdf · doi:10.48550/arxiv.2312.07092
We investigate the existence of normalized ground states for Schrödinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear δ-interactions at some of the vertices of the graph. For graphs with finitely many vertices, we show that ground states exist for every mass and every L2-subcritical power. For graphs with infinitely many vertices, we focus on periodic graphs and, in particular, on ℤ-periodic graphs and on a prototypical ℤ2-periodic graph, the two-dimensional square grid. We provide a set of results unravelling nontrivial threshold phenomena both on the mass and on the nonlinearity power, showing the strong dependence of the ground state problem on the interplay between the degree of periodicity of the graph, the total number of point defects and their dislocation in the graph.